A removable discontinuity is a break in a graph where the function is missing exactly one point, often called a hole. The curve may look smooth as it approaches the missing point from both sides, but the function value at that x-value is undefined or assigned a different value. These holes matter because limits can still exist even when the function itself is not defined at the point.
Recognizing removable discontinuities helps students separate the behavior of a graph from the value of the function at one input.
Many removable discontinuities appear in rational functions when the numerator and denominator share a common factor. Factoring and canceling the common factor reveals the simpler function that matches the graph everywhere except at the hole. The x-coordinate of the hole comes from the canceled factor, and the y-coordinate comes from substituting that x-value into the simplified expression.
The discontinuity can be removed by redefining the function so that its value at the hole equals the limit.
Understanding Calculus: Removable Discontinuities and Holes
Cancellation in algebra needs careful interpretation. When the same factor appears above and below a fraction bar, dividing by that factor gives a rule that produces the same outputs for every allowed input. It does not make the originally forbidden input allowed.
Division by zero occurred before any simplification, so the restriction remains part of the function's history. This is why a simplified formula can be useful for studying the nearby graph while still failing to give the complete definition of the original function.
Students often lose the hole by treating equivalent expressions as identical functions. They agree on their shared domain, but their domains may differ.
A hole is different from a vertical asymptote. Near a hole, the output values settle toward one ordinary finite height. Near a vertical asymptote, outputs grow without bound in size or fall without bound.
Factoring helps tell these cases apart. A denominator factor that cancels points to a possible hole. A denominator factor that remains after all possible cancellation points to an asymptote, unless another part of the expression changes the situation.
Make a sign chart or test values close to the excluded input when the graph is unclear. This prevents a small open circle from being confused with an unbounded branch.
Piecewise functions make the idea especially important. A function may follow one formula for inputs below a chosen number and another formula for inputs above it. If both formulas approach the same output, there is a single target value at that input.
The function can still have a hole if no rule includes the point, or if a separate rule assigns a different output there. On a graph, a filled dot shows the actual assigned value, while an open circle shows a value the nearby curve approaches but does not include.
A filled dot at a different height does not change the limit. Limits describe nearby behavior, not a single isolated output.
This distinction appears whenever a formula is used as a model rather than as a perfect description of reality. A measurement table may miss one time because a sensor failed, even though the values just before and after fit a smooth trend. In computer programs, an expression can fail at one input because it would require division by zero, even when a simplified calculation works for nearby inputs.
In calculus, repairing such a gap creates a continuous function, which is easier to integrate, differentiate, and use in later theorems. When solving problems, write excluded values before canceling, simplify only after recording them, then check the behavior from both sides.
Finally, separate three ideas in your work. State the nearby value, state whether the original function has a value there, and state what value would fill the gap.
Key Facts
- A removable discontinuity occurs when lim x->a f(x) exists but f(a) is undefined or not equal to the limit.
- A hole often comes from a common factor in a rational function, such as f(x) = (x - 2)(x + 3)/(x - 2).
- After canceling the common factor, f(x) = x + 3 for x != 2, so the graph follows y = x + 3 except at x = 2.
- The x-coordinate of a hole is found by setting the canceled factor equal to zero.
- The y-coordinate of a hole is found by substituting the hole's x-value into the simplified function.
- To remove the discontinuity at x = a, define f(a) = lim x->a f(x).
Vocabulary
- Removable discontinuity
- A discontinuity where a function has a hole that can be fixed by defining or redefining the function at one point.
- Hole
- A missing point on a graph where the function is not defined even though nearby values follow a clear pattern.
- Limit
- The value that a function approaches as the input gets closer to a certain number.
- Rational function
- A function that can be written as a fraction of two polynomials.
- Common factor
- A factor that appears in both the numerator and denominator of an algebraic fraction.
Common Mistakes to Avoid
- Canceling terms instead of factors is wrong because only entire multiplied factors may be canceled from a fraction.
- Forgetting that the canceled x-value is still excluded is wrong because the original denominator was zero at that input.
- Calling every denominator zero an asymptote is wrong because a canceled factor creates a hole, while a noncanceled factor may create a vertical asymptote.
- Using the original formula to find the hole's y-coordinate is wrong because the original function is undefined there; use the simplified formula instead.
Practice Questions
- 1 For f(x) = (x - 4)(x + 1)/(x - 4), find the coordinates of the hole.
- 2 For g(x) = (x^2 - 9)/(x - 3), simplify the function, state the excluded x-value, and find lim x->3 g(x).
- 3 Explain why h(x) = (x - 2)/(x - 2) has a removable discontinuity at x = 2 even though its simplified expression is 1.